Theorem u4lemnob 655

Description: Lemma for non-tollens implication study.

Assertion

Ref	Expression
u4lemnob	((a →4 b)⊥ ∪ b) = ((a ∩ b⊥ ) ∪ b)

Proof of Theorem u4lemnob

Step	Hyp	Ref	Expression
1		u4lemanb 600	. . . 4 ((a →4 b) ∩ b⊥ ) = ((a⊥ ∪ b) ∩ b⊥ )
2		oran2 84	. . . . 5 (a⊥ ∪ b) = (a ∩ b⊥ )⊥
3	2	ran 71	. . . 4 ((a⊥ ∪ b) ∩ b⊥ ) = ((a ∩ b⊥ )⊥ ∩ b⊥ )
4	1, 3	ax-r2 35	. . 3 ((a →4 b) ∩ b⊥ ) = ((a ∩ b⊥ )⊥ ∩ b⊥ )
5		anor1 80	. . 3 ((a →4 b) ∩ b⊥ ) = ((a →4 b)⊥ ∪ b)⊥
6		anor3 82	. . 3 ((a ∩ b⊥ )⊥ ∩ b⊥ ) = ((a ∩ b⊥ ) ∪ b)⊥
7	4, 5, 6	3tr2 61	. 2 ((a →4 b)⊥ ∪ b)⊥ = ((a ∩ b⊥ ) ∪ b)⊥
8	7	con1 63	1 ((a →4 b)⊥ ∪ b) = ((a ∩ b⊥ ) ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 16

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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